Show that the distance between two nonparallel lines is given by $\frac{|(p_2-p_1)\cdot (a_1\times a_2)|}{|| a_2\times a_1||}$

Let $P_1, P_2$ be the two parallel planes passing through $l_1, l_2$, respectively. Then the vector 
\[n=a_1 \times a_2\]
is a normal vector for $P_1$ and $P_2$.  Notice that The distance between the two lines is the same as the distance between the two parallel planes $P_1, P_2$.  Since $p_1 \in P_1$ and $p_2 \in P_2$, the distance between the planes is the length of the orthogonal projection of the vector $(p_2-p_1)$ onto the normal vector $n$, i.e. 
\[\text{Distance}=|| \frac{(p_2-p_1)\cdot (a_1\times a_2)}{||a_1\times a_2||^2} (a_1\times a_2)||\]
\[=| \frac{(p_2-p_1)\cdot (a_1\times a_2)}{||a_1\times a_2||^2}|   || (a_1\times a_2)||\]
\[=\frac{|(p_2-p_1)\cdot (a_1\times a_2)|}{||a_1\times a_2||}.\]